x,y,z满足x+y+z=3,x^2+y^2+z^2=29,x^3+y^3+z^3=45,求x,y,z,x^4+y^4+
来源:学生作业帮 编辑:神马作文网作业帮 分类:数学作业 时间:2024/11/19 03:58:59
x,y,z满足x+y+z=3,x^2+y^2+z^2=29,x^3+y^3+z^3=45,求x,y,z,x^4+y^4+z^4等于几
x+y+z=3 (1)
x²+y²+z²=29 (2)
x³+y³+z³=45 (3)
--------------------------------------------------------------------------
由方程(1)得
x+y=3-z (4)
(x+y)²=(3-z)²
x²+y²+2xy=9-6z+z²
x²+y²+z²+2xy=9-6z+2z²
29+2xy=9-6z+2z²
xy=z²-3z-10 (5)
--------------------------------------------------------------------------
由方程(3)得
(x+y)(x²-xy+y²)+z³=45
代入(4)和(5)得
(3-z)(29-z²-z²+3z+10)+z³=45
(3-z)(39-2z²+3z)+z³=45
z³-3z²-10z+24=0
(z³-6z²+12z-8)+(3z²-12z+12)-10z+20=0
(z-2)³+3(z-2)²-10(z-2)=0
(z-2)[(z-2)²+3(z-2)-10]=0
(z-2)(z-2+5)(z-2-2)=0
(z-2)(z+3)(z-4)=0
z=2,-3,4
--------------------------------------------------------------------------
z=2时,xy=-12,x+y=1 解得 x=4,y=-3 或 x=-3,y=4
z=-3时,xy=8,x+y=6 解得 x=4,y=2 或 x=2,y=4
z=4时,xy=-6,x+y=-1 解得 x=2,y=-3 或 x=-3,y=2
原方程组的解是
x = 4,y = 2,z = -3
x = -3,y = 2,z = 4
x = 2,y = 4,z = -3
x = -3,y = 4,z = 2
x = 4,y = -3,z = 2
x = 2,y = -3,z = 4
--------------------------------------------------------------------------
x^4+y^4+z^4
=(x²+y²)²-2x²y²+z^4
=(29-z²)²-2(z²-3z-10)²+z^4
=12z³-36z²-120z+641
=12(z-2)(z-4)(z+3)+353
=353
x²+y²+z²=29 (2)
x³+y³+z³=45 (3)
--------------------------------------------------------------------------
由方程(1)得
x+y=3-z (4)
(x+y)²=(3-z)²
x²+y²+2xy=9-6z+z²
x²+y²+z²+2xy=9-6z+2z²
29+2xy=9-6z+2z²
xy=z²-3z-10 (5)
--------------------------------------------------------------------------
由方程(3)得
(x+y)(x²-xy+y²)+z³=45
代入(4)和(5)得
(3-z)(29-z²-z²+3z+10)+z³=45
(3-z)(39-2z²+3z)+z³=45
z³-3z²-10z+24=0
(z³-6z²+12z-8)+(3z²-12z+12)-10z+20=0
(z-2)³+3(z-2)²-10(z-2)=0
(z-2)[(z-2)²+3(z-2)-10]=0
(z-2)(z-2+5)(z-2-2)=0
(z-2)(z+3)(z-4)=0
z=2,-3,4
--------------------------------------------------------------------------
z=2时,xy=-12,x+y=1 解得 x=4,y=-3 或 x=-3,y=4
z=-3时,xy=8,x+y=6 解得 x=4,y=2 或 x=2,y=4
z=4时,xy=-6,x+y=-1 解得 x=2,y=-3 或 x=-3,y=2
原方程组的解是
x = 4,y = 2,z = -3
x = -3,y = 2,z = 4
x = 2,y = 4,z = -3
x = -3,y = 4,z = 2
x = 4,y = -3,z = 2
x = 2,y = -3,z = 4
--------------------------------------------------------------------------
x^4+y^4+z^4
=(x²+y²)²-2x²y²+z^4
=(29-z²)²-2(z²-3z-10)²+z^4
=12z³-36z²-120z+641
=12(z-2)(z-4)(z+3)+353
=353
x=y/z=z/3,x+y+z =12,求2x+3y+4z是多少,
试证明(x+y-2z)+(y+z-2x)+(z+x-2y)=3(x+y-2z)(y+z-2x)(z+x-2y)
已知有理数x、y、z满足条件|x-z-2|+(3x-6y-7)^2+|3y+3z-4|=0,求x、y、z
已知x,y,z满足方程组:4x-5y+2z=o;x+4y-3x=0,求x:y:z的值
x/2=y/3=z/5 x+3y-z/x-3y+z
如果|x+y+z-6|+|2x+3y-z-12|+|2x-y-z|=0求x,y,
已知3x-2y-5z=0,2x-5y+4z=0,且x,y,z均不为0,求3x*x+2y*y+5z*z/5x*x+y*y-
已知x:y:z=3:4:5,3x+2y-4z=18.求:x+y+z.
已知x/3=y/4=z/2,求x+2y-z/x+y+z=?
x/3=y/4=z/5,求x+y+z/3x-2y+z的值,
若x/3=y/4=z/5,求x+y+z/3x-2y+z的值.
若x:y:z=2:3:4,求x+y+z/x-2y+3z的值